Crossquartile

Crossquartile is a descriptive statistics concept that analyzes the relationship between two quantitative variables by categorizing observations into quartiles for each variable and examining their joint distribution. The approach focuses on how often high or low values in one variable align with high or low values in the other, providing a view of association that is robust to outliers and nonlinearity.

Calculation typically proceeds as follows. For variables X and Y, compute the quartiles of each. Assign each observation to an X-quartile (Q1 through Q4) and a Y-quartile (Q1 through Q4). Build a 4-by-4 contingency table of counts Nij for quartile i of X and quartile j of Y, then convert to probabilities Pij = Nij / n. Compute marginals pi = sum_j Pij and qj = sum_i Pij. A simple summary is the diagonal share D = sum_i Pii, representing the proportion of observations that fall into corresponding quartiles across the two variables (low-low and high-high pairings).

Interpretation centers on the degree of alignment between the two variables. A diagonal share larger than would be expected under independence (as measured against the product of marginals) suggests positive crossquartile association; a smaller value suggests weak or negative association. A chi-square–like statistic can be used to quantify deviations from independence, but crossquartile analysis is often employed as an exploratory tool and visualization method (for example, via a heat map of the contingency table).

Applications and limitations: Crossquartile is used in exploratory data analysis, data quality assessment, and nonparametric investigations, especially when relationships may be nonlinear or affected by outliers. Limitations include information loss from discretization, sensitivity to quartile cut points, and reduced power for small samples. Related concepts include cross-tabulation, quantile-based analysis, and Spearman correlation.